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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Ming Antu's infinite series expansion of trigonometric functions ： ウィキペディア英語版 Ming Antu's infinite series expansion of trigonometric functions Ming Antu's infinite series expansion of trigonometric functions. Ming Antu, a court mathematician of the Qing dynasty did extensive work on infinite series expansion of trigonometric functions in his masterpiece Geyuan Milv Jifa''(Quick Method of Dissecting the Circle and Determination of The Precise Ratio of the Circe)''. Ming Antu built geometrical models based on a major arc of a circle and nth dissection of the major arc. In Fig 1, ''AE'' is the major chord of arc ''ABCDE'', and ''AB'', ''BC'', ''CD'', ''DE'' are its nth equal segments. If chord ''AE'' = ''y'', chord ''AB'' = ''BC'' = ''CD'' = ''DE'' = ''x'', find chord y as infinite series expansion of chord ''x''. He studied the cases of ''n'' = 2, 3, 4, 5, 10, 100, 1000 and 10000 in great detail in vol 3 and vol 4 of ''Geyuan Milv Jifa''. ==Historical background== In 1701, French Jesuit missionary(Pierre Jartoux 1668-1720) came to China, he brought along three infinite series expansion of trigonometry functions by Isaac Newton and J. Gregory:〔He Shaodong, "A Key Problem in the Study of Infinite Series", in ''The Qing Dynasty, Studies in the History of Natural Sciences'' vol 6 No3 1989 pp 205–214〕 :$\backslash pi=3\backslash left(1+\backslash frac+\backslash frac+\backslash frac+\backslash cdots\backslash right)$ :$\backslash sin\; x=x-\backslash frac+\backslash frac-\backslash frac+\backslash cdots$ :$\backslash operatorname\; x=\backslash frac-\backslash frac+\backslash frac+\backslash cdots.$ These infinite series stirred up great interest among Chinese mathematicians,as calculation of with these "quick methods" involve only multiplication, addition or subtraction, much faster than classic Liu Hui's π algorithm which involves taking square roots. However, Jartoux did not bring along the method for deriving these infinite series.Ming Antu suspected that the westerner did not want to share their secret,hence he set to work on it, and spent on and off for thirty years and completed a manuscript Geyuan Milv Jifa, he created geometrical models for obtaining trigonometric infinite series, and not only found the method for deriving the above three infinite series, but also discovered six more infinite series. In the process, he discovered and applied Catalan number. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ming Antu's infinite series expansion of trigonometric functions」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.